A map $f:X \to Y$ is ring-like if for every point $x \in X$ and every pair $U$ and $V$ of open neighborhoods of $x$ and $f(x)$ respectively, there is an open $W$ such that $f(x) \in W \subseteq V$ and $f^{−1}(\partial W) \subseteq U$. Ring-like maps were introduced by V.V.Fedorchuk in Bicompacta with noncoinciding dimensionalities, Dokl. Akad. Nauk SSSR 182 (1968), 275-277.

A ring-like map between compact spaces does not increase dimension (see Proposition 1.8 of this article).

A map $f:X \to Y$ is ring-like if for every point $x \in X$ and every pair $U$ and $V$ of open neighborhoods of $x$ and $f(x)$ respectively, there is an open $W$ such that $f(x) \in W \subseteq V$ and $f^{−1}(\partial W) \subseteq U$. Ring-like maps were introduced by V. V. Fedorchuk in Bicompacta with noncoinciding dimensionalities, Dokl. Akad. Nauk SSSR 182 (1968), 275-277. Note that if $Y$ has a base of clopen subsets then any $f:X \to Y$ is ring-like.

A ring-like map between compact spaces does not increase dimension (see Proposition 1.8 of this article).